Zero-offset seismic traces (x,z=0,t)do not provide an accurate picture of the subsurface layers when there is a great deal of
structural complexity.
For example, Figure 1.18 shows that the apparent reflection point deduced from our
traces does not coincide with the actual reflection point in which the reflection energy originated.

Figure 1.18:
For a dipping layer,
projecting reflection energy directly to a depth of vt below a trace defines the
apparent reflection point, which is not the same as the actual reflection point.
Thus the stacked section
d(x,z=0,t) is not a good approximation to m(x,z)for complex structures
or layers with steep dip.
Here t is the 1-way reflection time.

The inability of
d(x,z=0,t) to represent the seismic
section
gets even worse when your reflector becomes more complex as shown in
Figure 1.19.
For example, the faults in the data have diffraction tails
which are collapsed in the migrated section.
To correct for these distortions we apply the
operation of migration to the zero-offset seismic data
to produce an image m(x,z) of the reflectivity section in (x,z).

Formally, if
is the forward modeling operator so
that

=

(1.11)

then
migration can be described as the first iterate of a steepest descent
method:

=

(1.12)

The migration algorithm will be explained in detail by other speakers.
In fact, migration of the prestack data (i.e., CMP traces)
is now commonly used today
to improve the imaging quality even more.
Question: Why is
such a good approximation to L-1?
Answer: If the data
roughly resemble m,
then this suggests that
acts almost
like an identity operator in mapping model space to data space.
Thus,
might act like an inverse operator.
Also,
is somewhat diagonally dominant
so that its inverse can be roughly
approximated by a weighted diagonal matrix.

Figure 1.19:
(Top) Poststack migrated image and (bottom)
stacked seismic section.
Note how the faults are more clearly delineated and the
diffraction frowns
are collapsed in the migrated section.
Data are computed for
the SEG/EAGE synthetic overthrust model.